- Email: [email protected]
The global theory of flows in networks
ADVANCESIN
APPLIED
MATHBMATICS 1, 158-181 (1980)
The Global Theory L. H.
of Flows in Networks HARPER
Dqmrtment of Mathematics, University of Calijornia, Riverside. California 92521
The soil in which the results of this paper germinated was a problem posed by G.-C. Rota [15]: Can Sperner’s theorem (for the lattice of subsets of an n-set) be extended to the lattice of partitions of an n-set? My approach to this problem did not answer Rota’s question, but did lead to other extensions of Spemer’s theorem and (as is often the case with good problems) to discoveries of independent interest. Canfield [2-51 has answered Rota’s question in the negative. Using sophisticated techniques from probability theory he showed that for sufficiently large n (near Avogadro’s number) there is a counterexample. My own positive results on analogs of Sperner’s theorem have been reported elsewhere [ 10, 111.In this paper I wish to present those “results of independent interest.” These are certain constructions on networks (“network” is used here in the Ford-Fulkerson sense [8]) which preserve the common capacity (the maximum flow allowed by the network) of the constituent networks. I have found it helpful to place these results in the framework of category theory [13]. Category theory embodies the global or holistic point of view in mathematics just as set theory embodies the local or atomistic point of view. Set theory emphasizes the internal structure of mathematical objects, while category theory emphasizes the relationships between such objects and studies operations which construct new objects (of the same kind) from old ones. This global point of view crops up frequently in the literature of combinatorics (many standard results involve a notion of symmetry, reduction, or product), but seldom has been developed systematically. In the course of this study of flows on networks, however, the global questions and technical problems became sufficiently complicated to boggle the mind. Category theory has been of real assistance to me in arriving at a satisfactory understanding of the global aspects of flows. Circa 1969 I discovered the pushout for bipartite networks (see Sections 1.0 and 1.2) and posed the question of the existence of pullbacks (since I 158 0196-8858/80/020158-24$05.OO/O Copyright Au Ii&t8
0 1980 by Academic Press. Inc. of reproduction in any form reserved.
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knew no category theory, the names were different). Construction of the pullback eluded me though, until Dennis Johnson of the Jet Propulsion Laboratory pointed out that these were categorical concepts and that the theory could be used to construct “pullbacks.” MacLane’s book and discussions with Johnson and my colleagues, David Rush and Richard Block, soon made it clear that category theory had some substantial contributions to make to my project on networks, e.g., (i) a definitive point of view, with a body of theory and examples to support it; (ii) a language in which to formulate problems and state results; and (iii) several relevant theorems. The seminal result in the global theory of flows on networks appears in a paper by Graham and myself [9]. It is shown there that in the question of a matching between partitions of an n-set into k and (k + 1) blocks (essentially Rota’s problem), which by the Phillip Hall condition is equivalent to inequalities determined by subsets of partitions, one need only consider those subsets which are unions of families of partitions having the same unordered partitions (compositions) of n into k and (k + 1) summands. In [IO] I showed that what we had was a morphism from the network of set partitions to that of numerical partitions. Since the latter network is much smaller, the problem had been reduced. The following questions then presented themselves: (i) Are there other “independent” morphisms, either on II,, the network of set partitions, or on P,,, the network of numerical partitions? An approach to this would be to analyze the Graham-Harper morphism, and try to extend it. (ii) If an independent morphism is found on III,, would it necessarily induce one on P,,, i.e., reduce the size of the problem even further? The answer to the first question seems to be that there are no other systematic morphisms. The Graham-Harper morphism is the coequalizer (see Section 1.2) of the induced action of the symmetric group on II,, and there are evidently no other symmetries. The examples of Spencer [lo] and Canfield show that the positive results on Rota’s problem for small n are accidental. The answer to the second question is positive. Though rendered irrelevant to Rota’s problem by the negative answer to the first, pushouts do exist in the category of bipartite networks (see Sections 1.Oand 1.1) and give insight into flows on bipartite networks.
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A directed graph, G, consists of a set V of vertices, a set E of edges, and a pair of functions, a+,g _ : E + V, which identify the head and tail end, respectively, of each edge. Note that every directed graph is the image of a functor whose domain is the diagram category
aand whose codomain is the category SET of (finite) sets. The obvious notion of morphism for directed graphs G and H is a pair of functions cp,: E, + E, and (pV: V, + V,, such that the diagrams z EG-
VG
commute. & and c&, are then the components of a natural transformation from the functor defining G to that defining H. Thus DIGRAPH, the category of directed graphs with these morphisms, is the functor category FUNCT (- => -, SET), A network, N, consists of a directed graph, G, and a capacity function Y: V + R * . The vertices of G will be partitioned into three sets, R, S, and T. Members of S will be called sources, those of T sinks, and those of R intermediate vertices. A flow on N will be a function f: V + W such that
(0 fora s E S,&+~+A4 I Zadc+,f(e) Ids); (ii) for all t E T, Za-ccj-,f(e) I Za+c,,-,f(e) I v(r); and (iii) for all r E R, Za-ceJ-J(e) = &+feje,f(e) s ~(6
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The quantity 7t-f) = Zsas@a-~e~~sf(e) - Za+ce,JIe)) is &led the due of the flow, J It follows from the definition of a flow that r(j) is nonnegative and equal to &,A&+cej-,f(4 - Z,-ce,,,f(e)). me q=W K(N) = max, r(j)is called the cupuci~ of the network. Given a network, N, it is our task to compute K(N) and find a flow, f, such that 7(j) = K(N). In order to facilitate our definition of morphism for networks, we make two simple observations. (i) If f: A + W is a real-valued function on a finite set A, it may be uniquely extended to an additive function (measure) on the power set of A. f: ??(A) + W is defined by f(C) = Z,,&(a) for all C c A. Also, any measure on a finite set determines a function by restriction, so point-functions and measures are equivalent. Many of the facts about flows on networks, such as the Hall condition, are most naturally stated in terms of measures; so we shall assume that capacities and flows on networks are measures on V and E, respectively. (ii) If 9: A + B, A and B being finite sets, and v is a measure on A, then + induces a measure, +(v), on B by +(v)(C) = v(q5- ‘(C)) for all C C B. Now, let M and N be two networks, and suppose that +: GM + GN is a homomorphism for the underlying directed graphs such that &,(SM) c S,, +(7’,) c TN and &,(R,,J c RN. If +‘y(v& = vN, then $Jwill be called capacity preseming. Note. (i) If +: G,+,+ GN is capacity preserving, then all members of v, - +(VM) must have capacity zero. Since vertices of capacity zero are irrelevant, we shall ignore them and consider $ to be an epimorphism. (ii) If (p: G,,, + GN is capacity preserving, and f is a flow on iW, then r&(j) is a flow on N having the same value. Thus K(M) I K(N). It would be nice to have additional conditions on $J:GM + G, which guarantee equality between K(M) and K(N). This would hold if I$, as a function taking measures on E,,,, to measures on EN, had a right inverse which preserves flows and their values. A sufficient and in some sense necessary condition for this is the following: For an edge e E E,,,, if 8 +(e) # 3 -(e) let the subnetwork M, of M have vertices S, = ~:ca~(e))~~~d~~l(~+(e)), R, = 0, if a+(e) = a-(e) (i.e., e is a hop) let = = &*(3,(e)). Also let E= = {e’ E Enr: t&(e)) = e} akd v,(x) 2 v~(x)~v,,&,(x)) if x E R, u S, u T,. A flow on M, for which each of the upper bounds in the definition is achieved is called a normuIized flow. If e is not a loop, then M, is a bipartite graph and a normalized flow on iU, has value 1. If e is a loop, any flow on M, is a circulation, and a normalized flow is a maximum circulation.
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FUNDAMENTAL LEMMA. Zf for all edges e E EN, M, has a normalized flow, then 4: A4 + N has a right inverse which preserves flows and their values. Proof. We consider C& as a linear transformation taking functions on E,,., to functions on EN. Its right inverse, pE, will be a linear transformation taking functions on EN to functions on EM. For e E E,,, let x =: EN + R be
the indicator function of e, i.e., x,(x)
= 1 = 0
ifx=e otherwise.
Also, if f, is the normalized flow on I$, extend it to all of EM by letting it take value 0 on edges not in M,. Then since the x =‘s, e E EN, form a basis for all functions on EN, we may define ps, the right inverse of &, by pso( .) = f, (clearly (p,(f,) = x e, so it is a right inverse). Similarly, pv which takes functions of V, to functions on V, by
Pv(Xa)(x) = 3
d-4
if x E +‘(a)
dG’(a))
otherwise
0
is a right inverse for I#+ Now p = (pv, pE) constitutes a right inverse for + = (+“, +s). pE preserves flows since for any flow f on N and x E V,, 2 a+(+-
k(f)(e)
= a+#I=xf(+Ae))
-f&e)
vhf(x) I v (+,(x)) VNGtJYW = vhf(x)N
The other inequalities and equalities follow similarly, and p also preserves the values of these flows. Note. There are two conditions under which a bipartite network is easily seen to have a normalized flow: (i) The network is regular, i.e., if vx E A, a(x) = I{ e E E : a-(e) = x}j, and Vy E B, /3(y) = I{e E E : a+(e) = y}l, then v(x)/a(x) is constant on A as is v(y)/@(y) on B; and (ii) The network is complete; i.e., every vertex of A is connected to every vertex of B.
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A pair (+, p), + being determined by a graph homomorphism which is capacity preserving and p being a right inverse for +, is called a normal morphism. EXAMPLES
(0) All edgeswithout arrows are directed upward. t,
1’1
5
5
Circled vertices of M in the same circle are mapped to one vertex of N. (i) The notion of normal morphism was abstracted from [9], where M is the network whose vertices are the partitions of an n-set. S,,, is the discrete partition (having n blocks) and TM the indiscrete partition (one block). An edge e E EM will connect two partitions a -(e) and 8 +(e) such that two blocks of a-(e) can be joined to get a+(e). V,,.,(U)= 1. N’s vertices are (unordered) partitions of n, connected by an edge if one may be transformed into the other by simply adding a pair of summands. n!
+((a,. . . . , a/J) =
fi ai! (i!)” ’ i-l
where q is the number of summands of size i in (a,, . . . , uk). 9”((47 * * * , Bk}) = (pll, * * . , I&l) (IB,I 2 IBr+ll). Since for every edge e E EN, M, is a regular bipartite graph, M, has a normalized flow and the Fundamental Lemma applies.
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(ii) (due to M. B. Tomlinson [17]) Let M be any bipartite network and define functions g : A + 9(B) by g(x) = {a+(e) : g-(e) = x} and h : B + 9(A) by h(y) = {a-(e) : a+(e) = JJ}. The partitions of A and B determined by g and h, respectively, define a graph homomorphism which satisfies the Fundamental Lemma since the inverse image of each edge is a complete graph. This homomorphism may be just an isomorphism, but if \A) > 2”’ or IB ( > 214, it will be nontrivial. 1.0. LIMITS AND COLIMITS IN THE CATEGORY OF NETWORKS The category of networks and normal morphisms is a perfectly good category, but it lacks some of the universal constructions we would like it to have. The difficulties all trace back to the fact that the product measure is not a product of measure spacesin the categorical sense.The product of measurable spaces is uniquely determined (up to isomorphism), but there are generally many different measures on that space which make the projections measure-preserving. For example, the measures v and v’ defined on { 1,2} x { 1,2} by (x7 Y)
v(x, Y)
v’(x, Y)
both project onto the same marginals, but there is no measure-preserving function from either one to the other which factors projections. Ironically, it is this very nonuniqueness of product measure which gives rise to the theory of flows on networks. Since there may be many different measures on a product having specified projections, we may ask if there are any whose support is contained in a subset E of the product. If the sets are finite, E determines a bipartite network and such a measure is a flow. The nonuniqueness of product measure may be carried over to nonuniqueness of the pullback for normal morhpisms. Given a pullback diagram in the category of networks with normal morphisms, we may construct objects and morphisms in the category which have all the properties of pullbacks except for universality. This is not alarming, since in dealing with flows the emphasis should be on “construction” rather than “universal.” In fact it may even be the starting point for further investigation, just as the theory of flows starts with the nonuniversality of product measure.
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For colimits the situation seems more serious at first; because of the nonuniqueness of product measure, pushouts need not exist at all However, by altering the notion of morphism slightly, not only may we construct pushouts, but they are universal and thus actual colimits. The new notion of morphism which we shall use is that of a capacity-preserving graph homomorphism (p for which a flow-preserving right inverse exists. That is to say, C#Jis part of some normal morphism. Such a graph homomorphism will be called a flow morphism. To recap, a flow morphism + : it4 + N may be defined as (i) a graph epimorphism + : GM + GN, which is (ii) capacity preserving, +(Q,) = vN, and (iii) for all edges e E EN, M, has a normalized flow.
1.1.
“PULLBACKS”
The quotation marks denote the fact that limiting cones for pullback diugrums exist in the category of networks with normal morphisms, but they generally lack universality. The construction of the pullback cone is as follows: Let Nl
be a pullback diagram. Restrict it to the category of directed graphs (DIGRAPH) and we have the diagram
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Since DIGRAPH is a functor category over SET, it is complete and this diagram has a limit
Extend P to be a network by defining
where+,h) = cp2(u,)= u. With these definitions, IT, and r2 are capacity preserving since II; ‘(u,) = { ( ul, u2) : +,(u,) = Mu,) = 4 and
= -44 Y(+-yu) v(u) = v(q), since & is capacity preserving. Also (II,‘( II,‘(&)) is isomorphic to and II;’ 64;’ (4, %-’ (b)) as a graph and the measures on II;’ differ only by multiplicative constants from those on +;‘(a) and (p;‘(b). Thus the right inverses of s, and rITzare defined by the corresponding parts
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of the right inverses for C#J, and c$,, respectively, and because of this their compositions commute (to give a right inverse for $3,- rIT1= &“TJ. As mentioned before, the nonuniversality of product measure extends to nonuniversality of the above “pullback.” Any such “pullback,” P’, which makes
commute, will, when restricted to the underlying bipartite graph GPj,factor through GP. There will be a unique graph homomorphism + such that
commutes. There will be a unique measure on GP which will make C#J capacity preserving, and will also make ?T,and 7r2capacity preserving. n, and TV will have right inverses define by C#I- T;-’ and + a;- ‘, respectively. However, + will not generally have a right inverse, and so these pullbacks are not universal, even collectively.
1.2. PIJSHOIJTS
AND
COEQUALIZERS
As we mentioned in Section 1.O,the pushout diagram
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will not generally have a limiting cone. However, if we replace “normal morphism” by “flow morphism,” then a limiting cone may be constructed as follows: We again restrict the diagram
to its graphical component
which, by the cocompleteness of DIGRAPH, has a limiting cone
RQ
= smallest partition of RM larger than both
{+;‘(a,) : a, E R,}
and
{+,‘(a,)
: a2 E R,};
Se and TQ are similarly defined; Ep = {(a, b) : 3e E EM, a-(e) E a and a+(e) E b}, a-(~, b) = a and a+(~, b) = b. u, and u2 are defined by ui(u’)
= equivalence class of Gi- ‘(a’),
i = 1,2.
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The unique measure which makes u, and u2 capacity preserving is the measure of M restricted to the partition Ve. Thus (I, and uz are capacitypreserving graph homomorphisms and it remains to show that they have right inverses. By the Fundamental Lemma, we need only show that for each (a, 6) E Ee, (q-‘(a), u;‘(b)), i = 1,2, h as a normalized flow. Actually, it suffices to show that (a, b), as a subgraph of M, has a normalized flow. To see that it does, let Fa,bbe the set of all functions f : E,, b) -+ lR+ such that
F,, 6 is a compact set, so there exists f, E F,, b which minimizes
2 f(e) a+(e)Eb m(f) = max %f(b)
Clearly
*
m(fo) 2 1 and (a, b) will have a normalized flow if and only if
Nfo) = 1.
If for all bi E Bi, i = 1, 2, we let
Z: Qf)
= max
f(e)
a+(e)=b v(b’)
b’Eb,Cb
’
then m(j) = maxi m,,(j), since the bi partition b. Furthermore for all E 5, b and, m,,(pi * (Pi(f)) < nz,(j), because pi takes f E Fa,b, Pi ’ WI each edge of iVi to a normalized flow. Equality will hold iff
2
f(e)
a+(qsb v(b')
is constant on bi. Now if (a, 6) has no normalized flow, then m(fo) > 1 and there must exist b’, b” such that
2
a+ce)=b
Z
fd4
v(b’)
<
a+(e)-b"
v(b”)
fob9 = 4fo).
But then repeated applications of pi . r#+will change f. to f such that m(j) < m(fo). The number of repeated applications will be the length of
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the shortest path from b’ to 6” such that each consecutive pair lies in the same bi. A similar technique applies if e is a loop. Coequalizers may be constructed by the same process.
2.0. CONCLUSIONS AND COMMENTS
If I may be allowed a personal note here, my experience in writing this paper has been unique among my twenty-five or so research articles. Most of the others have been started and completed in a few months or a couple of years at most. The work on this one has extended over ten years, some sections having been rewritten on the order of fifty times. Mistakes that only become apparent when I tried to fill in details were responsible for many of these false starts, but also there were improvements that became apparent in the same way, particularly as I learned more category theory. For instance, most versions were written using bipartite networks rather than the more general setting used here. It was apparent that some generalization was possible, but several attempts to write the paper in greater generality ended in disaster; so I decided to play it safe. When a satisfactory version on bipartite networks was finally produced, I realized that the only fact about bipartite networks I had really used was that they were a functor category. Since the category of networks also had this property, it was only necessary to change the definitions and go through the manuscript to erase “bipartite” wherever it appeared, in order to extend the results.
2.1. THE SPERNER-ERD~S PROBLEM
As often happened in the course of this project, the preceding extension then led to other extensions: Flow morphisms were studied because they preserve flows, and therefore the Ford-Fulkerson capacity of the networks on which they act. But the Maxflow = Mincut Theorem suggeststhat they might preserve cuts also, and this turns out to be true. What other problems do they preserve? It would be nice if they preserved the Spemer problem; since this was the starting point for the study of flow morphisms, it would close the circle. It turns out that this is true and that it suggests many extensions of previous results on Sperner’s problem. A Spewer set of order k in P is a set S G P such that no more than k members of S lie on a chain in P. A Spemer set of order 1 is called an anti&in. The Sperner-Erdiis problem, given a weighted poset, P, and k E H+, is to find max sZ x =s~(x), over all Spemer sets of order k.
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A Sperner weight of order k on P is a measure w : P + W+ such that (i) for all x E P, o(x) I v(x), and (ii) for all chains C in P, ZxEc w(x)/Y(x)
I k.
Maximizing o(P) over all Spemer weights of order k is a linear programming problem which appears to extend the Spemer-Erdos problem; but since, as V. Strehl has shown, every Spemer-Erdos weight of order k is a convex combination of the weighted characteristic functions of Spemer sets of order k, they are actually equivalent problems. THEOREM. Zf P and Q are weighted posets and $J : P + Q is a jlowmorphism (on the acyclic directed graphs (Hasse diagram of P and Q)), then + preserues Sperner weights of order k and so the Sperner-Erdiis problems on P and Q are equiualent. LEMMA 1. Euery Sperner weight of order k, o, is a linear combination of weighted characteristic functions of antichains
where as 2 for all antichains, S, and
Proof. Induction on m = I{ x E P : w(x) > O}l gives proof. It is trivially true if m = 0. If m > 0, let S,,, = (x E P : w(x) > 0 and x maximal w. r. t. this property}. S, # $Jso (ys, = min{w(x)/v(x) : x E S} > 0, and S,,, is an antichain.
is a Spemer weight of order k’ = k - asm. Claim. wf=w~asvyxs, For if Z xEC~‘(~) I k?, C a saturated chain, then either C n {x E P : w(x) > O>is empty or it is nonempty and C n S,,, # 0. In either case 5 k - a. Also z ,..,+‘(x) m’ = 1(x E P : w’(x) > O}l
w=w’+
asmvxs .m= Zasvx s + a&vx s,.
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LEMMA2 (due to D. Kleitman [12] in the case V(X) E 1 ). If there is a flow morphism + : P + C, C a chain, then for any antichain A of P,
Proof: By induction on the rank of P, if r(P) = 0, the result is trivial. Assume it true for rank r 2 0 and that r(P) = r + 1. A,+ 1 = {x E A : r(x) = r + 1} may be removed from A and replaced by Ai = { x E P:r(x)=r,3y~A,+, and x < y }. Letting Ci be the element of rank i in C, A’ = (A - A,+l) u Ai is still an antichain, and
2 XEA
V(x) ‘(+(X>>
r+’ v(Ai) 2 is 1 p(ci)
ar-l r
dAi) 4
I ci)
4A’) 4
r
by the Fundamental Lemma and the Hall condition,
C,)
by the inductive hypothesis. Proof of the theorem. Let p be a right inverse of (p and let w’ be a Spemer weight of order k on Q. Then w’ = Z:x,Eew’(x’)xx, so
= &4x’Mx
.‘I
= sx&w xEF,(x,) &x*-WY Therefore for x E P,
d@‘)(X)= d+(x)) or
%4x) “&w)
t-44(x) = 4+(x)> %4x) ~&J(X>). Thus (i) 0 I &J’)(X) I +(x) for all x E P and (ii)
=
2
@‘cx’)
< k
X’Ecp(C)
“Q(X’)
-
for all chains, C, in P. Therefore p(w’) is a Spemer weight of order k.
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On the other hand, if w is a Spemer weight of order k on P, then
(9
~(4(x’)=
x 4x1 5 x 44 = Ql(x’> *G#l-‘(x’) xE$D‘(x’)
and (ii) if C’ is any chain Q, then x +(0)(x’) X’E C’ “Qb’)
=
2 1 X’EC’ “Qb’)
2 4x1 XE+-‘(,,)
the expansion of w being the convex combination guaranteed by Lemma 1. as 2 0 and Zsas I k. Therefore
5 2%
s I &
(by Lemma 2, since + restricted to +-‘(C’) is a flow morphism)
and we are done. What I called the “normalized flow property” (NFP; see [lo]) for a weighted poset P may now be succinctly defined as follows: P has NFP if there exists a flow morphism + : P + C, C a chain. Since the Ford-Fulkerson and Spemer-Erdos problems are trivial on a chain, a demonstration that a weighted poset has NFP solves both of these problems. Actually, Spemer’s original theorem and the extensions by Erdos, de Bruijn, and others-in fact all extensions of the Spemer-Erdiis type with one notable exception-follow from NFP. In the cases where NFP does hold, it is relatively easy to verify by global techniques such as coequalizers and the Product Theorem of [lo]. And not only do these techniques give immediate proofs of old results, but they point the way for new Spemertype theorems. More about this will be said later. For a time it seemed that NFP might also be necessary for extensions of Spemer’s theorem to reasonable families of posets; however, a recent result of Stanley’s shows this is not so. Stanley [16] has proved a Spemer-type theorem for the Brouhat ordering on quotients of Coxeter groups over hyperbolic subgroups. This has some interesting applications, including the solution of an old problem of Erdos and Moser. The Brouhat posets do not generally have NFP nor even any obvious flow morphisms. Stanley’s proof reduced the question of existence of the required matchings to Lefschetz’s
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Hard Theorem, a deep result in algebraic geometry. There are thus several indications that Stanley’s result is deeper than any extension of Spemer’s theorem which follows from NFP. One of my first results in the global theory of flows on networks was the Product Theorem of [lo]. It states that if Pi and Pz are weighted posets having NFP and such that the weights of the ranks of P, and Pz are 2-positive (logarithmically convex), then P, x Pz (the poset product with product weighting) also has NFP. The proof of this fact, as seen from the global point of view, is based on two facts: (i) Product in POSET, the category of partially ordered sets with order-preserving morphisms, is not a product in FLOW, the category of networks with flow morphisms, since the projections ri: P, X Pz + Pi, i = 1, 2, may map distinct pairs of comparable elements to the same point. However, it is still a functor from POSET’ x POSET’ + POSET’, POSET’ being the subcategory of FLOW whose networks are the Hasse diagrams of posets. That is, if +i: P, + Q, and &: Pz + Q, are flow morphisms, then r$i X &: P, X Pz -+ Q, X Q2 is also. This may be verified directly from the Fundamental Lemma. (ii) Now let Q, = C, and Q, = C,, the chains given in the hypothesis. Then the existence of 9’: C, x C, + C is implied by the 2 positivity of the weights and +’ 0 (+, X &): P, X P2 + C gives NFP for P, X P,. How can this result be extended? I conjectured that all geometric lattices should have NFP. Using the product theorem and Birkhoff representation, I could show that all modular geometric lattices had NFP. Also, all the lattices in Crapo’s catalog of irreducible geometric lattices with up to eight atoms had NFP. But I could not find a proof. Then in 1969 R. P. Dilworth sent me a counterexample! (See Fig. 1.) In his thesis C. Greene extended Dilworth’s example to a lattice of contractions of a planar graph for which even Spemer’s theorem does not hold [7]. A couple of years later, having learned a little category theory, it occurred to me that the Product Theorem might extend to pullbacks, product being a pullback over a terminal object. But this idea also foundered on Dilworth’s example, which turns out to be a pullback and actually very close to a product. (See Fig. 2.) The Dilworth-Greene example also has a simple representation as a pullback. These representations clarify the relationships of the examples to each other and to the Product Theorem. Last year a result of Peck [ 141opened up a new line of investigation: He showed that subrectangles of an n-dimensional rectangle have NFP. Realizing that subrectangles were just subintervals in the product of linear orders (the big rectangle), I investigated the functor Int: POSET + POSET. Under certain conditions Int extends to a functor on FLOW and I used this to extend Peck’s theorem to show that Int (P), P any modular geometric lattice, has NEP. This leads to a new question: Which functors
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FIG. 1. Hasse diagram of Dilworth’s
FIG. 2.
Representation
of Dilwortb’s
example.
example as a pullback.
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on POSET extend to functors on FLOW? This question seemslikely to be fruitful for extensions of Spemer’s theorem, and could not even be asked, much less answered, without category theory. 2.2. LOCAL TERMINALOBJECTS The category FLOW, whose objects are networks and whose morphisms are flow morphisms, seems to be the proper setting for a global theory of flows. The objects of any category may be quasi-ordered by saying N 5 A4 if there exists a morphism 9: M + N. From a quasi order one obtains an equivalence relation “- ” by defining M - N to mean M 5 N and N 5 M. The quasi order then gives a partial order on the equivalence classes of “- ” (See [I, p.201.) In an arbitrary category, the equivalence relation may be trivial, containing just one equivalence class, but in FLOW we have LEMMA. If there are flow morphisms $I,: A4 + N and Cp,:N + M, then M and N are isomophic. Proof: (p, * +i : M + M is a flow morphism and so must have a right inverse p . 641* $4 - P = k. But since the graph of M is finite and a function of a finite set to itself which is onto must be one-to-one, p = (&. cp,)-’ and (p, * +, is an isomorphism. Furthermore, (p . $2) . 9, = p . 642* (PII = iNTso (pi has a left inverse and must be an isomoxphism. Thus the equivalence relation in FLOW is just an isomorphism. As a category and as a quasi order, FLOW is not connected. If 9: M + N then V(S,) = Y(S~), V(TM) = V(TN) and the capacities of M and N must be the same. However, even if these conditions are satisfied by a pair of networks, they may not be in the same component of FLOW. LEMMA. Zf M, and M, are in the same component of FLOW, then there exists a network N such that
Proof: Since M, and M2 are in the same component, there must exist morphisms %,,.,j= 1, . . . , n, and networks N,, . . . , N,, such that N, =
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,..., n-l.Let Ml, Nn=MZand~j:Nj+Nj+,or+d:Nj+,+Nj,j=O n be minimal for such a sequence. Claim: n I 2. For if not, it could be shortened by taking compositions or pushouts or both. If n = 2 but
then we again use the pushout to obtain the desired form. THEOREM. Every component of FLOW, as a full subcategory, has a terminal object. Proof. The objects of FLOW are constructed from finite graphs. Since every flow morphism C#K A4 + N which is not an isomorphism must reduce the number of vertices in the graph, any chain
of nontrivial flow morphisms must eventually terminate. For each object N let T(N) be any object which terminates a maximal sequence starting from N. If N, and N, are in the same component of FLOW, then by the previous lemma T(N,) and T(N,) give a network N such that
But by the assumed extremality of T(N,) and T(NJ, T(N,) - N - T(N&. By the definition of T(N) there is a flow morphism Cp:N + T(N). If cp’:N + T(N) were a second such flow morphism, then the coequalizer of N 2 T(N) would contradict the extremality of T(N). Thus T(N) is a terminal object for the component of N.
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2.3. THE
COMPLEXITY
OF
FLOW MORPHISMS
In the Introduction we motivated the investigation of normal morphisms by observing that if we wish to compute the capacity of a network M which is the domain of a normal morphism Cp:M + N, then we need only compute the capacity of N. N may be much smaller than M, thus saving a lot of work. Suppose that we wish to compute the capacity of M but we do not know a priori whether M is the domain of any nontrivial normal morphisms. How could we determine this? The brute force method of answering this question is based upon the observation that every morphism +: M + N determines a partition of V. Conversely, given such a partition, we can construct a bipartite network N having the partition as its vertex set. A pair of such vertices will have an edge between them if any member of one is connected to any member of the other. vN will be vM restricted to blocks. With N so defined, then, the inclusion map Cp:M+ N is a capacity-preserving graph homomorphism. If every pair of blocks connected by an edge in N has a normalized flow, then, by the Fundamental Lemma, $I will be a normal morphism. Computing normal morphisms by this method would be terribly expensive, however. Unless some more efficient general algorithm of computing them can be found, we must be content with having them in those special cases where they are easily computed. It is really not clear, at this point, what the relationship between these notions of morphism and algorithms for computing capacity is. Another fact which complicates the picture here is that the “size” of a network is not just the number of vertices and edges it has, but the number of symbols required to represent it. We have shown that symmetries, morphisms whose domain and range are identical, will induce morphisms whose range is smaller via the coequalizer-smaller, that is, in the senseof having fewer vertices and edges. The proof for the existence of coequalizers, that they are inherited from SET, becomes an algorithm for constructing them, given such an algorithm in SET. The standard representation of coequalizers in SET, as equivalence classes of certain equivalence relations, is costly to compute and the end result is cumbersome, containing all the elements of the original set. Fortunately, though, limits are only determined up to isomorphism, and in many casesthere are less expensive representations. This is the case in the Graham-Harper example, where the coequalizer of the induced action of the symmetric group on partitions of an n-set is represented very nicely by partitions of n. This example brings out the importance of the complexity of all limits in SET and FLOW. Another interesting example in FLOW is T(N), the local terminal object for each network N. We could compute T(N) by computing all nontrivial flow morphisms on N, using the brute force method. The complexity of
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this task is exponential, but in certain cases it has been vastly simplified: The verifications of the normalized flow property for weighted and graded posets given in [lo, 111may be regarded as such, where T(N) is a chain with appropriate weights. I conjecture, however, that the question of whether a given network has a nontrivial flow morphism is NE-complete (see [ 181). 2.4. PULLBACKS AGAIN When normal morphisms failed to have pushouts, we were able to regain them by altering our notion of morphism slightly (from normal morphism to flow morphism). We still have a problem with pullbacks of flow morphisms, so one might ask if further small alterations on the category could relieve that too. This appears unlikely for the following reason: If pullbacks did exist, then each component of FLOW, having a terminal object, would have equalizers. (See [13, p.72, problem lo]. Actually, by Corollary 1, p. 109, the category would be complete for finite limits.) But since equalizers in SET are not generally surjective, our morphisms could no longer be determined by graph homomorphisms. Giving up graph homomorphisms as part of the structure of morphisms for networks would also require giving up the graphical structure of networks. But “networks” without graphs would be difficult to represent and it seems doubtful that there would be an effective way to compute the capacity of such a “network.” There are several other ways in which the present global theory of flows on bipartite networks might possibly be extended: (0) Capacities on edges. (i) The essenceof the notion of morphism for these structures is that it preserves flows and their values. In this paper we have guaranteed this by mapping vertices to vertices and edges to edges in underlying graphs. Is it possible, however, to map edges to vertices and vertices to edges and still maintain the essential structure? Such a notion would generalize the standard reductions of the capacity problem for networks having multiple sources and terminals to those having only one of each and for networks having vertex and edge capacities to those having only edge capacities or only vertex capacities. Of course a general theory is not needed for these reductions and nontrivial examples would be required to justify the effort. (ii) There should be a useful theory of continuous networks; e.g., let A and B be copies of [0, 11,the unit interval, with v, and v, Bore1 measures. If E is any measurable subset of the product A X' B, what is -f(E) over
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all nonnegative measures with support in E such that (1) vx G A, (2) VI’
c B,
f(X x B) c Q(X),
f(x x Y) c Yg(Y)?
Is there an effective way to approximate maxf(E), and is there an analog of the Hall condition? Evidently, these continuous analogs have been “much discussed but little developed.” My guess is that the cause of this state of affairs is the technical difficulty of answering the questions and the lack of good problems to motivate the effort. I suggest that the insight provided by our global theory of flows on networks, together with the possibility of applying new results to resolve continuous and asymptotic analogs of Rota’s problem, will break through the logjam.
1. G. BJRKHOFF, “Lattice Theory,” 3rd ed., Ame.r. Math. Sot. Colloquium Publications, Vol. XXV, Providence, R I., 1967. 2. E. R. CANPIBLD,Application of the Berry-Es&n inequality to combinatorial estimates, J. Combinatorial T?mny, Ser. A., 28 (1980). 17-25. 3. E. R. CANFIELD,Central and local limit theorems for the coefficients of polynomials of binomial type, J. Combinatorial Theory, Ser. A 23 (1977). 275-290. 4. E. R CANPIBLD,On the location of the maximum Stirling number (s) of the second kind, Stuaks Appl. Math 59 (1978), 83-93. 5. E. R. CANFIELD,On a problem of Rota, Aduames in Math. 29 (1978), l-10, 6. R. P. DIL.WORTH, private communication. 7. R P. DILWORTH AND C. GRFJBNE,A counterexample to the generalization of Spemer’s theorem, J. Combatoriai Theory 10 (1971), 18-21. “Flows in Networkx”, Van Nostrand, Princeton, N. J., 8. L. R. FORDAND D. R FIJLKJBWN, 1962. 9. RGAND L. H. HARPER,Some results on matching in bipartite graphs, SIAM J. Appl. Math. 17 (1%9), 1017-1022. 10. L. H. HARPER,The morphology of partiaUy ordered sets, J. Combinatoriaf m, Ser. B 17 (1974), 44-58.
11. L. H. H-RR, A Spemer theorem for the interval order of a projective geometry, J. Combinatorial Theoty, Ser. A. 28 (1980), 82-88. 12. D. J. KLBITUN, On an extremal property of antichains in partial orders: The LYM property and some of its implications and applications, in “Combinatorics” (M. Hall and J. H. van Lint, Eds.), 77-90, Math. Centre Tracts, Vol. 55, Math. Centre, Amsterdam, (1974). 13. S. MACLANE,“Categories for the Working Mathematician,” Springer-Verlag, New York, 1971. 14. G. W. Peck, Maximum antichains of rectangular arrays, J. Combinatorial Z%eov, Ser. A. 27 (1979), 397-400.
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15. G.-C. Ron, Research problem: A generalization of Spemer’s theorem, J. Combinaforiaf 77leoly 2 (1967). 104. 16. R P. STANLEY,Weyl groups, the hard Lefschetz theorem, and the Spemer property, preprint. 17. M. B. TOMUNSON,private communication. 18. M. R. GAREYAND D. S. JOHNSON“Computers and Intractability: A Guide to the Theory of NP-Completeness,” Freeman, San Fran&w, 1979.
APPLIED
MATHBMATICS 1, 158-181 (1980)
The Global Theory L. H.
of Flows in Networks HARPER
Dqmrtment of Mathematics, University of Calijornia, Riverside. California 92521
The soil in which the results of this paper germinated was a problem posed by G.-C. Rota [15]: Can Sperner’s theorem (for the lattice of subsets of an n-set) be extended to the lattice of partitions of an n-set? My approach to this problem did not answer Rota’s question, but did lead to other extensions of Spemer’s theorem and (as is often the case with good problems) to discoveries of independent interest. Canfield [2-51 has answered Rota’s question in the negative. Using sophisticated techniques from probability theory he showed that for sufficiently large n (near Avogadro’s number) there is a counterexample. My own positive results on analogs of Sperner’s theorem have been reported elsewhere [ 10, 111.In this paper I wish to present those “results of independent interest.” These are certain constructions on networks (“network” is used here in the Ford-Fulkerson sense [8]) which preserve the common capacity (the maximum flow allowed by the network) of the constituent networks. I have found it helpful to place these results in the framework of category theory [13]. Category theory embodies the global or holistic point of view in mathematics just as set theory embodies the local or atomistic point of view. Set theory emphasizes the internal structure of mathematical objects, while category theory emphasizes the relationships between such objects and studies operations which construct new objects (of the same kind) from old ones. This global point of view crops up frequently in the literature of combinatorics (many standard results involve a notion of symmetry, reduction, or product), but seldom has been developed systematically. In the course of this study of flows on networks, however, the global questions and technical problems became sufficiently complicated to boggle the mind. Category theory has been of real assistance to me in arriving at a satisfactory understanding of the global aspects of flows. Circa 1969 I discovered the pushout for bipartite networks (see Sections 1.0 and 1.2) and posed the question of the existence of pullbacks (since I 158 0196-8858/80/020158-24$05.OO/O Copyright Au Ii&t8
0 1980 by Academic Press. Inc. of reproduction in any form reserved.
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knew no category theory, the names were different). Construction of the pullback eluded me though, until Dennis Johnson of the Jet Propulsion Laboratory pointed out that these were categorical concepts and that the theory could be used to construct “pullbacks.” MacLane’s book and discussions with Johnson and my colleagues, David Rush and Richard Block, soon made it clear that category theory had some substantial contributions to make to my project on networks, e.g., (i) a definitive point of view, with a body of theory and examples to support it; (ii) a language in which to formulate problems and state results; and (iii) several relevant theorems. The seminal result in the global theory of flows on networks appears in a paper by Graham and myself [9]. It is shown there that in the question of a matching between partitions of an n-set into k and (k + 1) blocks (essentially Rota’s problem), which by the Phillip Hall condition is equivalent to inequalities determined by subsets of partitions, one need only consider those subsets which are unions of families of partitions having the same unordered partitions (compositions) of n into k and (k + 1) summands. In [IO] I showed that what we had was a morphism from the network of set partitions to that of numerical partitions. Since the latter network is much smaller, the problem had been reduced. The following questions then presented themselves: (i) Are there other “independent” morphisms, either on II,, the network of set partitions, or on P,,, the network of numerical partitions? An approach to this would be to analyze the Graham-Harper morphism, and try to extend it. (ii) If an independent morphism is found on III,, would it necessarily induce one on P,,, i.e., reduce the size of the problem even further? The answer to the first question seems to be that there are no other systematic morphisms. The Graham-Harper morphism is the coequalizer (see Section 1.2) of the induced action of the symmetric group on II,, and there are evidently no other symmetries. The examples of Spencer [lo] and Canfield show that the positive results on Rota’s problem for small n are accidental. The answer to the second question is positive. Though rendered irrelevant to Rota’s problem by the negative answer to the first, pushouts do exist in the category of bipartite networks (see Sections 1.Oand 1.1) and give insight into flows on bipartite networks.
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A directed graph, G, consists of a set V of vertices, a set E of edges, and a pair of functions, a+,g _ : E + V, which identify the head and tail end, respectively, of each edge. Note that every directed graph is the image of a functor whose domain is the diagram category
aand whose codomain is the category SET of (finite) sets. The obvious notion of morphism for directed graphs G and H is a pair of functions cp,: E, + E, and (pV: V, + V,, such that the diagrams z EG-
VG
commute. & and c&, are then the components of a natural transformation from the functor defining G to that defining H. Thus DIGRAPH, the category of directed graphs with these morphisms, is the functor category FUNCT (- => -, SET), A network, N, consists of a directed graph, G, and a capacity function Y: V + R * . The vertices of G will be partitioned into three sets, R, S, and T. Members of S will be called sources, those of T sinks, and those of R intermediate vertices. A flow on N will be a function f: V + W such that
(0 fora s E S,&+~+A4 I Zadc+,f(e) Ids); (ii) for all t E T, Za-ccj-,f(e) I Za+c,,-,f(e) I v(r); and (iii) for all r E R, Za-ceJ-J(e) = &+feje,f(e) s ~(6
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The quantity 7t-f) = Zsas@a-~e~~sf(e) - Za+ce,JIe)) is &led the due of the flow, J It follows from the definition of a flow that r(j) is nonnegative and equal to &,A&+cej-,f(4 - Z,-ce,,,f(e)). me q=W K(N) = max, r(j)is called the cupuci~ of the network. Given a network, N, it is our task to compute K(N) and find a flow, f, such that 7(j) = K(N). In order to facilitate our definition of morphism for networks, we make two simple observations. (i) If f: A + W is a real-valued function on a finite set A, it may be uniquely extended to an additive function (measure) on the power set of A. f: ??(A) + W is defined by f(C) = Z,,&(a) for all C c A. Also, any measure on a finite set determines a function by restriction, so point-functions and measures are equivalent. Many of the facts about flows on networks, such as the Hall condition, are most naturally stated in terms of measures; so we shall assume that capacities and flows on networks are measures on V and E, respectively. (ii) If 9: A + B, A and B being finite sets, and v is a measure on A, then + induces a measure, +(v), on B by +(v)(C) = v(q5- ‘(C)) for all C C B. Now, let M and N be two networks, and suppose that +: GM + GN is a homomorphism for the underlying directed graphs such that &,(SM) c S,, +(7’,) c TN and &,(R,,J c RN. If +‘y(v& = vN, then $Jwill be called capacity preseming. Note. (i) If +: G,+,+ GN is capacity preserving, then all members of v, - +(VM) must have capacity zero. Since vertices of capacity zero are irrelevant, we shall ignore them and consider $ to be an epimorphism. (ii) If (p: G,,, + GN is capacity preserving, and f is a flow on iW, then r&(j) is a flow on N having the same value. Thus K(M) I K(N). It would be nice to have additional conditions on $J:GM + G, which guarantee equality between K(M) and K(N). This would hold if I$, as a function taking measures on E,,,, to measures on EN, had a right inverse which preserves flows and their values. A sufficient and in some sense necessary condition for this is the following: For an edge e E E,,,, if 8 +(e) # 3 -(e) let the subnetwork M, of M have vertices S, = ~:ca~(e))~~~d~~l(~+(e)), R, = 0, if a+(e) = a-(e) (i.e., e is a hop) let = = &*(3,(e)). Also let E= = {e’ E Enr: t&(e)) = e} akd v,(x) 2 v~(x)~v,,&,(x)) if x E R, u S, u T,. A flow on M, for which each of the upper bounds in the definition is achieved is called a normuIized flow. If e is not a loop, then M, is a bipartite graph and a normalized flow on iU, has value 1. If e is a loop, any flow on M, is a circulation, and a normalized flow is a maximum circulation.
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FUNDAMENTAL LEMMA. Zf for all edges e E EN, M, has a normalized flow, then 4: A4 + N has a right inverse which preserves flows and their values. Proof. We consider C& as a linear transformation taking functions on E,,., to functions on EN. Its right inverse, pE, will be a linear transformation taking functions on EN to functions on EM. For e E E,,, let x =: EN + R be
the indicator function of e, i.e., x,(x)
= 1 = 0
ifx=e otherwise.
Also, if f, is the normalized flow on I$, extend it to all of EM by letting it take value 0 on edges not in M,. Then since the x =‘s, e E EN, form a basis for all functions on EN, we may define ps, the right inverse of &, by pso( .) = f, (clearly (p,(f,) = x e, so it is a right inverse). Similarly, pv which takes functions of V, to functions on V, by
Pv(Xa)(x) = 3
d-4
if x E +‘(a)
dG’(a))
otherwise
0
is a right inverse for I#+ Now p = (pv, pE) constitutes a right inverse for + = (+“, +s). pE preserves flows since for any flow f on N and x E V,, 2 a+(+-
k(f)(e)
= a+#I=xf(+Ae))
-f&e)
vhf(x) I v (+,(x)) VNGtJYW = vhf(x)N
The other inequalities and equalities follow similarly, and p also preserves the values of these flows. Note. There are two conditions under which a bipartite network is easily seen to have a normalized flow: (i) The network is regular, i.e., if vx E A, a(x) = I{ e E E : a-(e) = x}j, and Vy E B, /3(y) = I{e E E : a+(e) = y}l, then v(x)/a(x) is constant on A as is v(y)/@(y) on B; and (ii) The network is complete; i.e., every vertex of A is connected to every vertex of B.
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A pair (+, p), + being determined by a graph homomorphism which is capacity preserving and p being a right inverse for +, is called a normal morphism. EXAMPLES
(0) All edgeswithout arrows are directed upward. t,
1’1
5
5
Circled vertices of M in the same circle are mapped to one vertex of N. (i) The notion of normal morphism was abstracted from [9], where M is the network whose vertices are the partitions of an n-set. S,,, is the discrete partition (having n blocks) and TM the indiscrete partition (one block). An edge e E EM will connect two partitions a -(e) and 8 +(e) such that two blocks of a-(e) can be joined to get a+(e). V,,.,(U)= 1. N’s vertices are (unordered) partitions of n, connected by an edge if one may be transformed into the other by simply adding a pair of summands. n!
+((a,. . . . , a/J) =
fi ai! (i!)” ’ i-l
where q is the number of summands of size i in (a,, . . . , uk). 9”((47 * * * , Bk}) = (pll, * * . , I&l) (IB,I 2 IBr+ll). Since for every edge e E EN, M, is a regular bipartite graph, M, has a normalized flow and the Fundamental Lemma applies.
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(ii) (due to M. B. Tomlinson [17]) Let M be any bipartite network and define functions g : A + 9(B) by g(x) = {a+(e) : g-(e) = x} and h : B + 9(A) by h(y) = {a-(e) : a+(e) = JJ}. The partitions of A and B determined by g and h, respectively, define a graph homomorphism which satisfies the Fundamental Lemma since the inverse image of each edge is a complete graph. This homomorphism may be just an isomorphism, but if \A) > 2”’ or IB ( > 214, it will be nontrivial. 1.0. LIMITS AND COLIMITS IN THE CATEGORY OF NETWORKS The category of networks and normal morphisms is a perfectly good category, but it lacks some of the universal constructions we would like it to have. The difficulties all trace back to the fact that the product measure is not a product of measure spacesin the categorical sense.The product of measurable spaces is uniquely determined (up to isomorphism), but there are generally many different measures on that space which make the projections measure-preserving. For example, the measures v and v’ defined on { 1,2} x { 1,2} by (x7 Y)
v(x, Y)
v’(x, Y)
both project onto the same marginals, but there is no measure-preserving function from either one to the other which factors projections. Ironically, it is this very nonuniqueness of product measure which gives rise to the theory of flows on networks. Since there may be many different measures on a product having specified projections, we may ask if there are any whose support is contained in a subset E of the product. If the sets are finite, E determines a bipartite network and such a measure is a flow. The nonuniqueness of product measure may be carried over to nonuniqueness of the pullback for normal morhpisms. Given a pullback diagram in the category of networks with normal morphisms, we may construct objects and morphisms in the category which have all the properties of pullbacks except for universality. This is not alarming, since in dealing with flows the emphasis should be on “construction” rather than “universal.” In fact it may even be the starting point for further investigation, just as the theory of flows starts with the nonuniversality of product measure.
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For colimits the situation seems more serious at first; because of the nonuniqueness of product measure, pushouts need not exist at all However, by altering the notion of morphism slightly, not only may we construct pushouts, but they are universal and thus actual colimits. The new notion of morphism which we shall use is that of a capacity-preserving graph homomorphism (p for which a flow-preserving right inverse exists. That is to say, C#Jis part of some normal morphism. Such a graph homomorphism will be called a flow morphism. To recap, a flow morphism + : it4 + N may be defined as (i) a graph epimorphism + : GM + GN, which is (ii) capacity preserving, +(Q,) = vN, and (iii) for all edges e E EN, M, has a normalized flow.
1.1.
“PULLBACKS”
The quotation marks denote the fact that limiting cones for pullback diugrums exist in the category of networks with normal morphisms, but they generally lack universality. The construction of the pullback cone is as follows: Let Nl
be a pullback diagram. Restrict it to the category of directed graphs (DIGRAPH) and we have the diagram
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HARPER
Since DIGRAPH is a functor category over SET, it is complete and this diagram has a limit
Extend P to be a network by defining
where+,h) = cp2(u,)= u. With these definitions, IT, and r2 are capacity preserving since II; ‘(u,) = { ( ul, u2) : +,(u,) = Mu,) = 4 and
= -44 Y(+-yu) v(u) = v(q), since & is capacity preserving. Also (II,‘( II,‘(&)) is isomorphic to and II;’ 64;’ (4, %-’ (b)) as a graph and the measures on II;’ differ only by multiplicative constants from those on +;‘(a) and (p;‘(b). Thus the right inverses of s, and rITzare defined by the corresponding parts
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of the right inverses for C#J, and c$,, respectively, and because of this their compositions commute (to give a right inverse for $3,- rIT1= &“TJ. As mentioned before, the nonuniversality of product measure extends to nonuniversality of the above “pullback.” Any such “pullback,” P’, which makes
commute, will, when restricted to the underlying bipartite graph GPj,factor through GP. There will be a unique graph homomorphism + such that
commutes. There will be a unique measure on GP which will make C#J capacity preserving, and will also make ?T,and 7r2capacity preserving. n, and TV will have right inverses define by C#I- T;-’ and + a;- ‘, respectively. However, + will not generally have a right inverse, and so these pullbacks are not universal, even collectively.
1.2. PIJSHOIJTS
AND
COEQUALIZERS
As we mentioned in Section 1.O,the pushout diagram
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will not generally have a limiting cone. However, if we replace “normal morphism” by “flow morphism,” then a limiting cone may be constructed as follows: We again restrict the diagram
to its graphical component
which, by the cocompleteness of DIGRAPH, has a limiting cone
RQ
= smallest partition of RM larger than both
{+;‘(a,) : a, E R,}
and
{+,‘(a,)
: a2 E R,};
Se and TQ are similarly defined; Ep = {(a, b) : 3e E EM, a-(e) E a and a+(e) E b}, a-(~, b) = a and a+(~, b) = b. u, and u2 are defined by ui(u’)
= equivalence class of Gi- ‘(a’),
i = 1,2.
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The unique measure which makes u, and u2 capacity preserving is the measure of M restricted to the partition Ve. Thus (I, and uz are capacitypreserving graph homomorphisms and it remains to show that they have right inverses. By the Fundamental Lemma, we need only show that for each (a, 6) E Ee, (q-‘(a), u;‘(b)), i = 1,2, h as a normalized flow. Actually, it suffices to show that (a, b), as a subgraph of M, has a normalized flow. To see that it does, let Fa,bbe the set of all functions f : E,, b) -+ lR+ such that
F,, 6 is a compact set, so there exists f, E F,, b which minimizes
2 f(e) a+(e)Eb m(f) = max %f(b)
Clearly
*
m(fo) 2 1 and (a, b) will have a normalized flow if and only if
Nfo) = 1.
If for all bi E Bi, i = 1, 2, we let
Z: Qf)
= max
f(e)
a+(e)=b v(b’)
b’Eb,Cb
’
then m(j) = maxi m,,(j), since the bi partition b. Furthermore for all E 5, b and, m,,(pi * (Pi(f)) < nz,(j), because pi takes f E Fa,b, Pi ’ WI each edge of iVi to a normalized flow. Equality will hold iff
2
f(e)
a+(qsb v(b')
is constant on bi. Now if (a, 6) has no normalized flow, then m(fo) > 1 and there must exist b’, b” such that
2
a+ce)=b
Z
fd4
v(b’)
<
a+(e)-b"
v(b”)
fob9 = 4fo).
But then repeated applications of pi . r#+will change f. to f such that m(j) < m(fo). The number of repeated applications will be the length of
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the shortest path from b’ to 6” such that each consecutive pair lies in the same bi. A similar technique applies if e is a loop. Coequalizers may be constructed by the same process.
2.0. CONCLUSIONS AND COMMENTS
If I may be allowed a personal note here, my experience in writing this paper has been unique among my twenty-five or so research articles. Most of the others have been started and completed in a few months or a couple of years at most. The work on this one has extended over ten years, some sections having been rewritten on the order of fifty times. Mistakes that only become apparent when I tried to fill in details were responsible for many of these false starts, but also there were improvements that became apparent in the same way, particularly as I learned more category theory. For instance, most versions were written using bipartite networks rather than the more general setting used here. It was apparent that some generalization was possible, but several attempts to write the paper in greater generality ended in disaster; so I decided to play it safe. When a satisfactory version on bipartite networks was finally produced, I realized that the only fact about bipartite networks I had really used was that they were a functor category. Since the category of networks also had this property, it was only necessary to change the definitions and go through the manuscript to erase “bipartite” wherever it appeared, in order to extend the results.
2.1. THE SPERNER-ERD~S PROBLEM
As often happened in the course of this project, the preceding extension then led to other extensions: Flow morphisms were studied because they preserve flows, and therefore the Ford-Fulkerson capacity of the networks on which they act. But the Maxflow = Mincut Theorem suggeststhat they might preserve cuts also, and this turns out to be true. What other problems do they preserve? It would be nice if they preserved the Spemer problem; since this was the starting point for the study of flow morphisms, it would close the circle. It turns out that this is true and that it suggests many extensions of previous results on Sperner’s problem. A Spewer set of order k in P is a set S G P such that no more than k members of S lie on a chain in P. A Spemer set of order 1 is called an anti&in. The Sperner-Erdiis problem, given a weighted poset, P, and k E H+, is to find max sZ x =s~(x), over all Spemer sets of order k.
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A Sperner weight of order k on P is a measure w : P + W+ such that (i) for all x E P, o(x) I v(x), and (ii) for all chains C in P, ZxEc w(x)/Y(x)
I k.
Maximizing o(P) over all Spemer weights of order k is a linear programming problem which appears to extend the Spemer-Erdos problem; but since, as V. Strehl has shown, every Spemer-Erdos weight of order k is a convex combination of the weighted characteristic functions of Spemer sets of order k, they are actually equivalent problems. THEOREM. Zf P and Q are weighted posets and $J : P + Q is a jlowmorphism (on the acyclic directed graphs (Hasse diagram of P and Q)), then + preserues Sperner weights of order k and so the Sperner-Erdiis problems on P and Q are equiualent. LEMMA 1. Euery Sperner weight of order k, o, is a linear combination of weighted characteristic functions of antichains
where as 2 for all antichains, S, and
Proof. Induction on m = I{ x E P : w(x) > O}l gives proof. It is trivially true if m = 0. If m > 0, let S,,, = (x E P : w(x) > 0 and x maximal w. r. t. this property}. S, # $Jso (ys, = min{w(x)/v(x) : x E S} > 0, and S,,, is an antichain.
is a Spemer weight of order k’ = k - asm. Claim. wf=w~asvyxs, For if Z xEC~‘(~) I k?, C a saturated chain, then either C n {x E P : w(x) > O>is empty or it is nonempty and C n S,,, # 0. In either case 5 k - a. Also z ,..,+‘(x) m’ = 1(x E P : w’(x) > O}l
w=w’+
asmvxs .m= Zasvx s + a&vx s,.
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LEMMA2 (due to D. Kleitman [12] in the case V(X) E 1 ). If there is a flow morphism + : P + C, C a chain, then for any antichain A of P,
Proof: By induction on the rank of P, if r(P) = 0, the result is trivial. Assume it true for rank r 2 0 and that r(P) = r + 1. A,+ 1 = {x E A : r(x) = r + 1} may be removed from A and replaced by Ai = { x E P:r(x)=r,3y~A,+, and x < y }. Letting Ci be the element of rank i in C, A’ = (A - A,+l) u Ai is still an antichain, and
2 XEA
V(x) ‘(+(X>>
r+’ v(Ai) 2 is 1 p(ci)
ar-l r
dAi) 4
I ci)
4A’) 4
r
by the Fundamental Lemma and the Hall condition,
C,)
by the inductive hypothesis. Proof of the theorem. Let p be a right inverse of (p and let w’ be a Spemer weight of order k on Q. Then w’ = Z:x,Eew’(x’)xx, so
= &4x’Mx
.‘I
= sx&w xEF,(x,) &x*-WY Therefore for x E P,
d@‘)(X)= d+(x)) or
%4x) “&w)
t-44(x) = 4+(x)> %4x) ~&J(X>). Thus (i) 0 I &J’)(X) I +(x) for all x E P and (ii)
=
2
@‘cx’)
< k
X’Ecp(C)
“Q(X’)
-
for all chains, C, in P. Therefore p(w’) is a Spemer weight of order k.
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On the other hand, if w is a Spemer weight of order k on P, then
(9
~(4(x’)=
x 4x1 5 x 44 = Ql(x’> *G#l-‘(x’) xE$D‘(x’)
and (ii) if C’ is any chain Q, then x +(0)(x’) X’E C’ “Qb’)
=
2 1 X’EC’ “Qb’)
2 4x1 XE+-‘(,,)
the expansion of w being the convex combination guaranteed by Lemma 1. as 2 0 and Zsas I k. Therefore
5 2%
s I &
(by Lemma 2, since + restricted to +-‘(C’) is a flow morphism)
and we are done. What I called the “normalized flow property” (NFP; see [lo]) for a weighted poset P may now be succinctly defined as follows: P has NFP if there exists a flow morphism + : P + C, C a chain. Since the Ford-Fulkerson and Spemer-Erdos problems are trivial on a chain, a demonstration that a weighted poset has NFP solves both of these problems. Actually, Spemer’s original theorem and the extensions by Erdos, de Bruijn, and others-in fact all extensions of the Spemer-Erdiis type with one notable exception-follow from NFP. In the cases where NFP does hold, it is relatively easy to verify by global techniques such as coequalizers and the Product Theorem of [lo]. And not only do these techniques give immediate proofs of old results, but they point the way for new Spemertype theorems. More about this will be said later. For a time it seemed that NFP might also be necessary for extensions of Spemer’s theorem to reasonable families of posets; however, a recent result of Stanley’s shows this is not so. Stanley [16] has proved a Spemer-type theorem for the Brouhat ordering on quotients of Coxeter groups over hyperbolic subgroups. This has some interesting applications, including the solution of an old problem of Erdos and Moser. The Brouhat posets do not generally have NFP nor even any obvious flow morphisms. Stanley’s proof reduced the question of existence of the required matchings to Lefschetz’s
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Hard Theorem, a deep result in algebraic geometry. There are thus several indications that Stanley’s result is deeper than any extension of Spemer’s theorem which follows from NFP. One of my first results in the global theory of flows on networks was the Product Theorem of [lo]. It states that if Pi and Pz are weighted posets having NFP and such that the weights of the ranks of P, and Pz are 2-positive (logarithmically convex), then P, x Pz (the poset product with product weighting) also has NFP. The proof of this fact, as seen from the global point of view, is based on two facts: (i) Product in POSET, the category of partially ordered sets with order-preserving morphisms, is not a product in FLOW, the category of networks with flow morphisms, since the projections ri: P, X Pz + Pi, i = 1, 2, may map distinct pairs of comparable elements to the same point. However, it is still a functor from POSET’ x POSET’ + POSET’, POSET’ being the subcategory of FLOW whose networks are the Hasse diagrams of posets. That is, if +i: P, + Q, and &: Pz + Q, are flow morphisms, then r$i X &: P, X Pz -+ Q, X Q2 is also. This may be verified directly from the Fundamental Lemma. (ii) Now let Q, = C, and Q, = C,, the chains given in the hypothesis. Then the existence of 9’: C, x C, + C is implied by the 2 positivity of the weights and +’ 0 (+, X &): P, X P2 + C gives NFP for P, X P,. How can this result be extended? I conjectured that all geometric lattices should have NFP. Using the product theorem and Birkhoff representation, I could show that all modular geometric lattices had NFP. Also, all the lattices in Crapo’s catalog of irreducible geometric lattices with up to eight atoms had NFP. But I could not find a proof. Then in 1969 R. P. Dilworth sent me a counterexample! (See Fig. 1.) In his thesis C. Greene extended Dilworth’s example to a lattice of contractions of a planar graph for which even Spemer’s theorem does not hold [7]. A couple of years later, having learned a little category theory, it occurred to me that the Product Theorem might extend to pullbacks, product being a pullback over a terminal object. But this idea also foundered on Dilworth’s example, which turns out to be a pullback and actually very close to a product. (See Fig. 2.) The Dilworth-Greene example also has a simple representation as a pullback. These representations clarify the relationships of the examples to each other and to the Product Theorem. Last year a result of Peck [ 141opened up a new line of investigation: He showed that subrectangles of an n-dimensional rectangle have NFP. Realizing that subrectangles were just subintervals in the product of linear orders (the big rectangle), I investigated the functor Int: POSET + POSET. Under certain conditions Int extends to a functor on FLOW and I used this to extend Peck’s theorem to show that Int (P), P any modular geometric lattice, has NEP. This leads to a new question: Which functors
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FIG. 1. Hasse diagram of Dilworth’s
FIG. 2.
Representation
of Dilwortb’s
example.
example as a pullback.
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on POSET extend to functors on FLOW? This question seemslikely to be fruitful for extensions of Spemer’s theorem, and could not even be asked, much less answered, without category theory. 2.2. LOCAL TERMINALOBJECTS The category FLOW, whose objects are networks and whose morphisms are flow morphisms, seems to be the proper setting for a global theory of flows. The objects of any category may be quasi-ordered by saying N 5 A4 if there exists a morphism 9: M + N. From a quasi order one obtains an equivalence relation “- ” by defining M - N to mean M 5 N and N 5 M. The quasi order then gives a partial order on the equivalence classes of “- ” (See [I, p.201.) In an arbitrary category, the equivalence relation may be trivial, containing just one equivalence class, but in FLOW we have LEMMA. If there are flow morphisms $I,: A4 + N and Cp,:N + M, then M and N are isomophic. Proof: (p, * +i : M + M is a flow morphism and so must have a right inverse p . 641* $4 - P = k. But since the graph of M is finite and a function of a finite set to itself which is onto must be one-to-one, p = (&. cp,)-’ and (p, * +, is an isomorphism. Furthermore, (p . $2) . 9, = p . 642* (PII = iNTso (pi has a left inverse and must be an isomoxphism. Thus the equivalence relation in FLOW is just an isomorphism. As a category and as a quasi order, FLOW is not connected. If 9: M + N then V(S,) = Y(S~), V(TM) = V(TN) and the capacities of M and N must be the same. However, even if these conditions are satisfied by a pair of networks, they may not be in the same component of FLOW. LEMMA. Zf M, and M, are in the same component of FLOW, then there exists a network N such that
Proof: Since M, and M2 are in the same component, there must exist morphisms %,,.,j= 1, . . . , n, and networks N,, . . . , N,, such that N, =
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,..., n-l.Let Ml, Nn=MZand~j:Nj+Nj+,or+d:Nj+,+Nj,j=O n be minimal for such a sequence. Claim: n I 2. For if not, it could be shortened by taking compositions or pushouts or both. If n = 2 but
then we again use the pushout to obtain the desired form. THEOREM. Every component of FLOW, as a full subcategory, has a terminal object. Proof. The objects of FLOW are constructed from finite graphs. Since every flow morphism C#K A4 + N which is not an isomorphism must reduce the number of vertices in the graph, any chain
of nontrivial flow morphisms must eventually terminate. For each object N let T(N) be any object which terminates a maximal sequence starting from N. If N, and N, are in the same component of FLOW, then by the previous lemma T(N,) and T(N,) give a network N such that
But by the assumed extremality of T(N,) and T(NJ, T(N,) - N - T(N&. By the definition of T(N) there is a flow morphism Cp:N + T(N). If cp’:N + T(N) were a second such flow morphism, then the coequalizer of N 2 T(N) would contradict the extremality of T(N). Thus T(N) is a terminal object for the component of N.
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2.3. THE
COMPLEXITY
OF
FLOW MORPHISMS
In the Introduction we motivated the investigation of normal morphisms by observing that if we wish to compute the capacity of a network M which is the domain of a normal morphism Cp:M + N, then we need only compute the capacity of N. N may be much smaller than M, thus saving a lot of work. Suppose that we wish to compute the capacity of M but we do not know a priori whether M is the domain of any nontrivial normal morphisms. How could we determine this? The brute force method of answering this question is based upon the observation that every morphism +: M + N determines a partition of V. Conversely, given such a partition, we can construct a bipartite network N having the partition as its vertex set. A pair of such vertices will have an edge between them if any member of one is connected to any member of the other. vN will be vM restricted to blocks. With N so defined, then, the inclusion map Cp:M+ N is a capacity-preserving graph homomorphism. If every pair of blocks connected by an edge in N has a normalized flow, then, by the Fundamental Lemma, $I will be a normal morphism. Computing normal morphisms by this method would be terribly expensive, however. Unless some more efficient general algorithm of computing them can be found, we must be content with having them in those special cases where they are easily computed. It is really not clear, at this point, what the relationship between these notions of morphism and algorithms for computing capacity is. Another fact which complicates the picture here is that the “size” of a network is not just the number of vertices and edges it has, but the number of symbols required to represent it. We have shown that symmetries, morphisms whose domain and range are identical, will induce morphisms whose range is smaller via the coequalizer-smaller, that is, in the senseof having fewer vertices and edges. The proof for the existence of coequalizers, that they are inherited from SET, becomes an algorithm for constructing them, given such an algorithm in SET. The standard representation of coequalizers in SET, as equivalence classes of certain equivalence relations, is costly to compute and the end result is cumbersome, containing all the elements of the original set. Fortunately, though, limits are only determined up to isomorphism, and in many casesthere are less expensive representations. This is the case in the Graham-Harper example, where the coequalizer of the induced action of the symmetric group on partitions of an n-set is represented very nicely by partitions of n. This example brings out the importance of the complexity of all limits in SET and FLOW. Another interesting example in FLOW is T(N), the local terminal object for each network N. We could compute T(N) by computing all nontrivial flow morphisms on N, using the brute force method. The complexity of
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this task is exponential, but in certain cases it has been vastly simplified: The verifications of the normalized flow property for weighted and graded posets given in [lo, 111may be regarded as such, where T(N) is a chain with appropriate weights. I conjecture, however, that the question of whether a given network has a nontrivial flow morphism is NE-complete (see [ 181). 2.4. PULLBACKS AGAIN When normal morphisms failed to have pushouts, we were able to regain them by altering our notion of morphism slightly (from normal morphism to flow morphism). We still have a problem with pullbacks of flow morphisms, so one might ask if further small alterations on the category could relieve that too. This appears unlikely for the following reason: If pullbacks did exist, then each component of FLOW, having a terminal object, would have equalizers. (See [13, p.72, problem lo]. Actually, by Corollary 1, p. 109, the category would be complete for finite limits.) But since equalizers in SET are not generally surjective, our morphisms could no longer be determined by graph homomorphisms. Giving up graph homomorphisms as part of the structure of morphisms for networks would also require giving up the graphical structure of networks. But “networks” without graphs would be difficult to represent and it seems doubtful that there would be an effective way to compute the capacity of such a “network.” There are several other ways in which the present global theory of flows on bipartite networks might possibly be extended: (0) Capacities on edges. (i) The essenceof the notion of morphism for these structures is that it preserves flows and their values. In this paper we have guaranteed this by mapping vertices to vertices and edges to edges in underlying graphs. Is it possible, however, to map edges to vertices and vertices to edges and still maintain the essential structure? Such a notion would generalize the standard reductions of the capacity problem for networks having multiple sources and terminals to those having only one of each and for networks having vertex and edge capacities to those having only edge capacities or only vertex capacities. Of course a general theory is not needed for these reductions and nontrivial examples would be required to justify the effort. (ii) There should be a useful theory of continuous networks; e.g., let A and B be copies of [0, 11,the unit interval, with v, and v, Bore1 measures. If E is any measurable subset of the product A X' B, what is -f(E) over
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all nonnegative measures with support in E such that (1) vx G A, (2) VI’
c B,
f(X x B) c Q(X),
f(x x Y) c Yg(Y)?
Is there an effective way to approximate maxf(E), and is there an analog of the Hall condition? Evidently, these continuous analogs have been “much discussed but little developed.” My guess is that the cause of this state of affairs is the technical difficulty of answering the questions and the lack of good problems to motivate the effort. I suggest that the insight provided by our global theory of flows on networks, together with the possibility of applying new results to resolve continuous and asymptotic analogs of Rota’s problem, will break through the logjam.
1. G. BJRKHOFF, “Lattice Theory,” 3rd ed., Ame.r. Math. Sot. Colloquium Publications, Vol. XXV, Providence, R I., 1967. 2. E. R. CANPIBLD,Application of the Berry-Es&n inequality to combinatorial estimates, J. Combinatorial T?mny, Ser. A., 28 (1980). 17-25. 3. E. R. CANFIELD,Central and local limit theorems for the coefficients of polynomials of binomial type, J. Combinatorial Theory, Ser. A 23 (1977). 275-290. 4. E. R CANPIBLD,On the location of the maximum Stirling number (s) of the second kind, Stuaks Appl. Math 59 (1978), 83-93. 5. E. R. CANFIELD,On a problem of Rota, Aduames in Math. 29 (1978), l-10, 6. R. P. DIL.WORTH, private communication. 7. R P. DILWORTH AND C. GRFJBNE,A counterexample to the generalization of Spemer’s theorem, J. Combatoriai Theory 10 (1971), 18-21. “Flows in Networkx”, Van Nostrand, Princeton, N. J., 8. L. R. FORDAND D. R FIJLKJBWN, 1962. 9. RGAND L. H. HARPER,Some results on matching in bipartite graphs, SIAM J. Appl. Math. 17 (1%9), 1017-1022. 10. L. H. HARPER,The morphology of partiaUy ordered sets, J. Combinatoriaf m, Ser. B 17 (1974), 44-58.
11. L. H. H-RR, A Spemer theorem for the interval order of a projective geometry, J. Combinatorial Theoty, Ser. A. 28 (1980), 82-88. 12. D. J. KLBITUN, On an extremal property of antichains in partial orders: The LYM property and some of its implications and applications, in “Combinatorics” (M. Hall and J. H. van Lint, Eds.), 77-90, Math. Centre Tracts, Vol. 55, Math. Centre, Amsterdam, (1974). 13. S. MACLANE,“Categories for the Working Mathematician,” Springer-Verlag, New York, 1971. 14. G. W. Peck, Maximum antichains of rectangular arrays, J. Combinatorial Z%eov, Ser. A. 27 (1979), 397-400.
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15. G.-C. Ron, Research problem: A generalization of Spemer’s theorem, J. Combinaforiaf 77leoly 2 (1967). 104. 16. R P. STANLEY,Weyl groups, the hard Lefschetz theorem, and the Spemer property, preprint. 17. M. B. TOMUNSON,private communication. 18. M. R. GAREYAND D. S. JOHNSON“Computers and Intractability: A Guide to the Theory of NP-Completeness,” Freeman, San Fran&w, 1979.